For n∈N,n≥2, prove that
n∑k=1(12k−1−12k)>2n3n+1.
Note that
1−12+13−14+15−16⋯+12n−1−12n
=1−12+13−14+15−16+⋯+12n−1−12n
=(1+13+15+⋯+12n−1)−(12+14+16+⋯+12n)
=(1+12+13+⋯+12n)−2(12+14+16+⋯+12n)
=(1+12+13+⋯+12n)−(1+12+13+⋯+1n)
=(1+12+13+⋯+1n+1n+1+1n+2+⋯+12n)−(1+12+13+⋯+1n)
=1n+1+1n+2+⋯+12n
Therefore
n∑k=1(12n−1−12n)=1n+1+1n+2+⋯+12n≥(n times⏞1+1+⋯+1)2n+1+n+2+⋯+2n(by the extended Cauchy-Schwarz inequality)=n2n2(n+1+2n)=2n3n+1(Q.E.D.)
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